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For more in the same genre, the "Project Mathematics!" videos from Tom Apostol's clan at Caltech are absolutely superb.

They cover material which is generally more "elementary" than calculus: the Pythagorean theorem, trigonometry, polynomials and so forth. Even so, when we screened several at MIT a couple years ago, the math majors sat back and asked, "Why didn't I think of it that way before?" It looks like they're available on DVD now, too, so I might finally be able to replace the VHS tapes I dubbed off the NASA channel ten years ago.

Jennifer, have you visited Pyracantha at Electron Blue - ? She's been teaching herself maths for a while now. You're on her blogroll too!

I applaud your self-education efforts, Jen, but "mathier" is not one of the things I think I should be. It makes a great deal of sense for you to be learning calculus with the work you do, but I can't really see it helping me in any concrete way. Maybe ignorance is bliss, but there are so many other subjects I'd really like to know way more about that flogging myself to learn math beyond Algebra II and Geometry (subjects that used to make me weep quite regularly in high school) seems (let me get my flame-retardant suit on here first) a waste of my time. Not that I don't understand the appeal of math--it's a great feeling when your equations balance and your theorems work--but personally, I find a great deal more satisfaction in a well-turned sentence or a beautiful piece of music.

I think the idea of innumerancy is something of a false analogy. Illiteracy makes life in the modern world extremely difficult: passing a driving test, getting directions, understanding instructions, even understanding the sell-by date on your milk. (With our culture becoming increasingly driven by visual media, even this isn't as true as previously, but I digress). And yet I (and many other people, I suspect) have never felt at a loss in my life because of my poor math skills, which cover balancing my checkbook and figuring percentages, except in science class. Even there, it didn't hinder my understanding of the concepts, and it weeded me out of any career in chemistry. Oh well.

Math is cool and interesting and useful and it's a fascinating way of explaining the world, but it's only one way. And it has varying degrees of usefulness, as does, say, a knowledge of chemistry. I don't have to have a knowledge of the math underlying the design of Brunelleschi's dome to understand what an engineering feat it was in its day, or to appreciate its beauty.

But don't let that deter your inner math nerd. Get your calculus geek on!

I'd like to hear whay 400 red blooded american mothers would have to say if it were their children had been infected with HIV,
I'm guessing they'd shout blue murder
And in true Republican style flatten Tripoli.

But hey these are not our children, right (or is it left). As for maths I understand other people's children are usually worth something several orders of magnitude less in the mother's eyes.
This would appear to hold true:
whether you are Palestinian, Israeli or Lebanese
whether you are Lybian, American (US) or Rumanian

But this is not a Cosmological constant
No, this is not even a Planck constant
This is a 'nurtured' and evolved constant

Have a great weekend, and don;t worry tooo much about being mathier, words are just letters used in complex formulae to have meaning in language!
And there are many languages and many alphabets, and maths (geometry, algebra, trigger...) is one set of languages spoken (common) to those living in the 'pocket' universe of maths. You know any calculator can perform equations, but electronic translators make a right hash of words, meaning, context and nuances.

Rocky & Rambo proved you don't need a dialogue to make a film interesting, and I think most people would have turned the volume down (or off) if Tom cruise would have spewed out too many mathematical equations in Top Gun or Mission Impossible.

I do not mean we should not encourage children (boys & girls) to be learned and develop an interest in science. But Maths is not TOE and the Universe is not composed of numbers, any more than it is composed of music, they are only a part of the composition, the vibrant & dynamic Universe.

Have a great weekend!

Very few people need calculus (or even algebra) in their daily lives. I think not understanding the beauty of, say, the Lagrange Inversion Theorem ( leaves one a little spiritually impoverished, but it's no big deal. Where most could really benefit is in being a little bit 'physicier', and in particular having a smidgen of that most essential scientific gift: the highly sensitive BS detector. One of the things that we were drilled in when I was doing my Physics degree (Imperial College London) was how to train oneself to have a fair idea of the range in which the outcome of an experiment should fall. This carries over into daily life: it enables one to sniff out bogus claims and skewed statistics. I can't count the number of times someone's fired off some random factoid, and I've thought, 'hmm, that can't be right, because for it to be true, such-and-such would have to be x, and that's clearly ridiculous." I've won nontrivial (heh) sums of money in bets from simply sticking to my guns and calling people out on their claims. Without the ability to critically examine the slew of junk science garbage and politicians' lies that assail us every day, the average person is like a drunk stumbling through a mugger-infested alley. Sooner or later he's going to get rolled.

David, I think the "BS detector" you're talking about has less to do with a particular discipline than the faculty of critical thinking, which isn't much taught in schools where test scores are all that matters. It's true that to be a critical thinker, it helps to have at least a smidgen of disciplines like math, statistics, and some kind of science, but skepticism is the primary ingredient, as is understanding how research and criticism work, and how to look things up. You can get all those skills in the humanities as well. Once one decides not to take everything one is fed at face value, whole new worlds open up.

Lee, you're absolutely right that a BS detector can be possessed by all, but I wonder to what extent the relativist doctrine of there being no 'privileged discourses' militates against its acquisition in Humanities disciplines. If you look at the root of the word 'skeptic', we see that its original meaning is to examine or consider. So many of us go through life without examining the world around us and, more importantly, our own outlooks and assumptions. One of the biggest boosts that a scientific education gives you is the art of self-examination. As Richard Feynman put it, scientists are really the only class of people who in their daily lives are trying to prove themselves wrong.

Yay for learning calculus! No, you don't need it for functional everyday applications, but as you know from your research into giants of physics, the development of calculus was a central achievement of the Enlightenment. We began to understand that physical events didn't happen because of some capricious deity but rather could be predicted with the proper mathematical tools. Not only will you be getting mathier, but you will be rounding out your education in Western cultural history.

When did it become "not okay to be illiterate." I've had several co-workers and a couple of bosses that were illiterate and they didn't think it a handicap. George Bush managed to become president without knowing what the grecians like to call themselves.

When you write documentation for software, keep in mind that your target audience is just going to call tech support to have them read the document over the phone and explain the process in one syllable words. ("Click left mouse button. No left. Towards your heart.")


Did you know that's what the 'grecians' like to call themselves? Didn't think so.

Knowing a certain amount of mathematics is an important (I almost wrote "essential") prerequisite for being an informed citizen of the modern world, if only because people with ulterior agendas manipulate numbers in order to lie. Unfortunately, the skills necessary to ferret out such lies, including a basic knowledge of statistics, do not really overlap with the categories of mathematics taught in our schools. Eit.

When Carl Sagan wrote up his "baloney detection kit", he included **misunderstanding of statistics** as a key item people should watch for. His example was that Dwight Eisenhower had once expressed astonishment that half of all Americans were of below-average intelligence; any **informed** citizen should know that the nature of the average is to lie in the middle. This particular symptom of innumeracy continues to enjoy a privileged life.

Once while flying from Tokyo to Chigaco, I was desperate enough for input that I actually opened the in-flight magazine. Among the first things I read was an article about a new publishing imprint being established at one of the major publishers, a line of books expressly intended "for the conservative reader". The executive they quoted about this daring venture said that it was really quite hard to understand why the "New York literary scene" neglected conservative readers, because studies show that roughly half of all people are politically conservative to some extent!

"Yes, dear publisher," I thought, "and half of all people score better than average at miniature golf."

I almost wrote a letter to the airline magazine pointing out the innumeracy of the executive, asking if that had been the best quote of the interview, wondering what such acumen implied about the new imprint's leadership, and delicately suggesting that a good logo for the imprint would be an ostrich burying its head in the sand. By the time my plane landed in Chicago, I had decided that snarkiness was not the better part of valor, and I kept quiet.

For examples of the ways bad math aids and abets a wide variety of scoundrels, check out Mark Chu-Carroll's blog, simply entitled "Good Math, Bad Math":

Also recommended are Keith Devlin's writings on mathematics, including his regular column "Devlin's Angle". There's lots of good stuff in those archives, including "Dear President Bush" (January 2003):

More directly germane to the discussions here is "Letter to a calculus student" (June 2006):

I suspect that inserting any more URLs will make the spam filter vomit my message, so I'll stop here.

I hope both the gentle readers here and the copyright police elsewhere will forgive my posting a relevant passage. The text is from a 1941 essay by Jorge Luis Borges; my translation is by Eliot Weinberger and appears on pages 208--209 of **Selected Non-Fictions**. Borges writes as follows:

**Let the People Think** is the title of a selection of essays by Bertrand Russell. Wells, in the book I outlined above, urges us to rethink the history of the world without geographical, economic or ethnic preferences; Russell also advises universality. In the third article, "Free Thought and Official Propaganda," he proposes that elementary schools teach the art of reading the newspaper with incredulity. I believe that this Socratic discipline would not be useless. Of the people I know, very few practice it at all. They let themselves be deceived by typographical or syntactical devices; they think that an event has occurred because it is printed in large black letters; they don't want to know that the statement "All the aggressor's attempts to advance beyond B have failed miserably" is merely a euphemism for admitting the loss of B. Even worse: they practice a kind of magic, and think that to express any fear is to collaborate with the enemy. . . . Russell proposes that the State attempt to immunize people against such deceptions and sophistries. For example, he suggests that students should study Napoleon's final defeats through the ostensibly triumphant bulletins in **Moniteur**. A typical assignment would be to read the history of the wars with France in English textbooks, and then to rewrite that history from the French point of view. Our own "nationalists" have already adopted that paradoxical method: they teach Argentine history from a Spanish viewpoint, if not Quechua or Querandi.

[end quoted material]

The relevance of mathematics to this effort should be obvious. Borges and Russell have already devised a homework assignment; I suggest a key textbook would be Darrell Huff's **How to Lie With Statistics**, first published in 1954 and still readily available and relevant.

In passing, one should note that **Selected Non-Fictions**, from which I copied the above passage, is an epidemic of examples illustrating how mysteries of mathematics provoked and stimulated one of the twentieth century's quintessential "literature people". The stories, essays and poetry which make so many people feel that Borges's not receiving a Nobel Prize was an injustice were informed as much by mathematics as by mythology, as much by Cantor's set theory as by **The Thousand and One Nights**. Indeed, Borges is the only source I know of who called Cantor's theory "heroic". . . .

All the way back in 1966, Vladimir Nabokov called the gulf between C. P. Snow's two cultures "a mere dimple of a ditch that a small frog could straddle." (The year before, Kurt Vonnegut had observed that those cultures could indeed "sweetly intertwine", doing so the most in the minds of science-fiction editors and anthologists.) The study of the "humanities" is only the inane rearrangement of grunts transcribed on paper if the student has not shared the experience of beauty; knowing that beauty is the essential ingredient and Prime Mover, it would be criminal to neglect science and mathematics as sources of poetry.

David, good point about "privileged discourses" but I've never taken that particular bit of humanities theory very seriously, myself, being a trained skeptic. :^) It's starting to wear out its welcome in the academy too, I think. I do wish statistics (or "Lies, damned lies, and statistics" as Benjamin Disraeli phrased it) were taught at least to all undergraduates. Realizing how easily manipulated statistics are was one of my most disillusioning moments, and there aren't enough people who do understand that.

Blake, the Borges quotation is wonderful, but it's just as easily read as referring to prose writers' tricks of the trade as to mathematics. Language is just as twisty as statistics in the right hands. And much as I've read Borges (which is nearly all of his fiction and some poetry), I've never thought of him as particularly mathematically influenced. That was news to me, and interesting, too. But I think some of this interpretation depends upon one's own point of view.

And I agree that science and math shouldn't be neglected as sources of beauty (see my earlier post on science and poetry on Jen's blog)--just that it's in the eye of the beholder.

How are "The Library of Babel" and "The Book of Sand" not ineluctably, inescapably mathematical in nature? Or, say, the story of the coin with only one side, which makes its holder King. . . .

I feel like I tried to make a point which has been thoroughly missed. If "language is just as twisty as statistics in the right hands," then doesn't that mean "statistics is twisty as hell"? Therefore, how can Russell's plan to have the State immunize its people against the evils of Government work without teaching those people the fallacies employed on both sides of the coin, with tricky words and also with tricky numbers? Here, the question is not beauty, but civic responsibility.

(For those wordanistas doing a "close reading" at home, yes, the juxtaposition of "State" and "Government" was meant to be mildly paradoxical, and the repetition of the coin image was deliberate.)

An odd thing has puzzled me for several years now. Did anyone else ever wonder why the same people --- English teachers --- are responsible for teaching both beauty and utility? It's true: we give these folks the task of instructing students in how to write a "persuasive essay" (three paragraphs plus one each for introduction and conclusion), and then we give the same individuals the mission to make kids respond to poetry. This sounds like a losing proposition to me. Writing clearly, cogently and perhaps persuasively is indeed an important task for later adult life, but should we entrust the instruction in that skill to **literature teachers**? The very thought makes my fingers tremble and the blood drain noticably from my face.

By contrast, the math teachers have it easy. We only expect them to teach utility. Students receive the answer "you'll need it when you grow up" whenever they ask why they must learn the material, an answer I **never** heard given when we read Madeleine L'Engle, Lois Lowry, Harper Lee, F. Scott Fitzgerald, Walt Whitman, William Shakespeare, Sophocles, Hermann Hesse or James Joyce. To be fair, I never heard "beauty" given as justification for any of those, either; occasionally, we were told about some measure of historical importance (then why did we never read Thomas Paine?), but most often, the justification began and ended with the exam. I shudder to contemplate the number of good books which have died in students' minds for just that reason. . . .

On those grounds, I suppose trying to teach the beauty of science and mathematics would not only be impractical, but also a double standard. Which leaves, regrettably, practical usefulness and civic responsibility as the primary reason we offload mathematics onto children for oh so many years.

"first principals" a very old concept. i was looking on the internet for rumored dvd(s) a series of lectures by r. feynman @ caltec. i read in a scijournal his estate and caltec were i legal dispute. but i digressed, i wanted these images to relieve my enjoyment of a phone call from the greatest physical chemist of the 20th century inviting me to hear df lecture. usually little room for df's students. feynman made physics understandable and fun. i tried to emulate df in my lectures but never came near, he was truly "one of a kind" to see his equations on his van is/was a real "crack-up"(old slang, a common term of that era.

I've argued about this on several forums (fora?) and most people seem to think I'm crazy. The Eisenhower anecdote is based on a fallacy. Half the american population is very obviously NOT below average intelligence, no matter how you define average, as mean or median or whatever. There are only about 100 IQs available from standard tests, correct? There are how many million Americans? So, each available IQ will apply, not to one person, but to a group, and average will be a very, very large group - millions I imagine. So, that's a large number disposed of already. Thus there aren't enough left for those below average to constitute half the total. Soemone who should know, told me that the Eisenhower anecdote is sort of an "arithmetical joke or sleight of hand, rather than a real assertion about the number of actual Americans with a given IQ score".

I'm having trouble following your argument. If you define "average" as the median, then half of any set will lie below the "average", by definition. In a Gaussian distribution, colloquially known as a bell-shaped curve, the median and the mean coincide, and half the items being measured will fall below the mean, too.

Let's say that the national distribution of IQ scores is a bell-shaped curve centered on 100. Sure, millions of people will have an IQ of 100 (the most common value, because it's at the top of the bell). Millions of others will have an IQ of 95, for example, although the total number with IQ 95 will be less than the total number with IQ 100. Add up all the people whose IQ is less than 100, and that'll be equal to the number of people whose IQ is greater than 100, because a bell-shaped curve is symmetric down the middle.

There's a slight issue with what you do with the people who score exactly 100, but with a Gaussian curve, what you'll find is that 1% of the people will score exactly 100, 49.5% will score below 100 and 49.5% will score above (the actual figures will depend upon how widely spread-out the bell curve is).

Ultimately, the reason this whole thing is a "joke" is because IQ is not a very good measure of intelligence, so we're really being more precise than we should -- using the heavy machinery of mathematics in the soft mud of everyday speech.

Well, this is extraordinary. Blake, you say you aren't following my argument, then you proceed to illustrate my point perfectly! You agree that millions of people will have an IQ of 100. That is my point, and the whole of my point. Half the population means just that - half the number of INDIVIDUALS. It may well be that the number below 100 IQ will be equal to the number above 100 IQ, but neither number will constitute half the population, will it? Half the population, plus or minus one, will have to be above or below a single solitary American person, which of course is quite unreal. To "isolate" that person, so to speak, you'd have to split that group scoring 100, and you can't, because there's no such thing as part of an IQ. Let me quote more from my expert source: "The average referred to does not relate to the people or objects measured to create the distribution, it refers simply to the value of the number at the point on the graph where the sample is exactly divided into two. To indicate the difference between the two concepts, it can be seen that the number which equates to the arithmetic mean may be a number whcich it is not actually possible for any given person to achieve on the measurement concerned (for example, it may be half way in between two test scores). The Eisenhower anecdote involves an elision between the people in the distribution (50% of Americans) and the arithmetic mean (their average score - the score which splits the observed distribution in half)."
My source, by the way, is the supervisory psychologist at the UK branch of MENSA. As for the statement that 1% of the people will score exactly 100, my understanding is that the range 90-110 covers no less than 50% of the people, which seems to me in my innocence to indicate that there will be more than 1% on 100. But even that 1% would suffice to prove my original contention.

People who make comments about George Bush's peanut-size brain typically make their judgments on two factors: 1) The President's lack of speaking skill, and 2) His positions with which the critic disagrees. I've known many highly intelligent people who had a hard time communicating, making the same sort of speaking errors for which Bush is famous. It seems that they think too far ahead and don't give themselves time to get their thoughts out. Mentally they are thinking of the next thought in their train of thought rather than concentrating on communicating. Those who stutter often possess extraordinary intelligence but have difficulty in connecting their brain with their tongue. Bill Walton, for example, has a high IQ and thinks logically but was such a bad stutterer at one time that he hardly talked. On the other hand, I've known people who were of average intelligence and possessed very limited skills at thinking logically; yet they were as glib as Bob Costas. I submit that publicly criticizing Bush for lack of intelligence reflects the commentator's lack of understanding. Don't forget that Bush did better in school than Gore. And a critical analysis of Gore's lecture known as "An Inconvenient Truth" shows a man who posssesses above average communication skills but limited powers of logical thinking. If the "scientists" who laud Gore's movie were not already committed to limiting human impact on the world, they would be severe critics of his presentation.

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    • Heavy G
      The perfect pick-me-up when gravity gets you down.
      2 oz Tequila
      2 oz Triple sec
      2 oz Rose's sweetened lime juice
      7-Up or Sprite
      Mix tequila, triple sec and lime juice in a shaker and pour into a margarita glass. (Salted rim and ice are optional.) Top off with 7-Up/Sprite and let the weight of the world lift off your shoulders.
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      1 oz dark rum
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      In a shaker half-filled with ice, combine the dark and light rum, Tia Maria, and cream. Shake well. Strain into an old fashioned glass almost filled with crushed ice. Dust with the nutmeg, and serve. Bongos optional.
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      Fill shot glass 3/4 with Triple Sec. Layer Bailey's on top. Drop Grenadine in center of shot; it should billow up like a mushroom cloud. Remember to "duck and cover."
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      Any mad scientist will tell you that flames make drinking more fun. What good is science if no one gets hurt?
      1 oz Midori melon liqueur
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      Warning: may result in amplified stimulated emission.
      1 oz Southern Comfort
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      7 oz orange juice
      Combine all liquor in a full glass of ice. Shake well. Garnish with orange and cherry. Serve to attractive target of choice.
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      3/4 oz Rum
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